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We study depth properties of a general class of random recursive trees where each node  i  attaches to the random node  \documentclass{article} \usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty} \begin{document} \begin{align*}\left\lfloor iX_i\right\rfloor\end{align*} \end{document} and  X  0 ,…, X   n   is a sequence of i.i.d. random variables taking values in [0,1). We call such trees scaled attachment random recursive trees (sarrt). We prove that the typical depth  D   n  , the maximum depth (or height)  H   n   and the minimum depth  M   n   of a sarrt are asymptotically given by  D   n   ∼μ ‐1  log  n ,  H   n   ∼ α max  log  n  and  M   n   ∼ α min  log  n  where μ,α max  and α min  are constants depending only on the distribution of  X  0  whenever  X  0  has a density. In particular, this gives a new elementary proof for the height of uniform random recursive trees  H   n   ∼  e log  n  that does not use branching random walks.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011

Depth Properties of scaled attachment random recursive trees